On the set of catenary degrees of finitely generated cancellative commutative monoids

The catenary degree of an element s of a cancellative commutative monoid S is a nonnegative integer measuring the distance between the irreducible factorizations of s. The catenary degree of the monoid S, defined as the supremum over all catenary degrees occurring in S, has been heavily studied as an invariant of nonunique factorization. In this paper, we investigate the set C(S) of catenary degrees achieved by elements of S as a factorization invariant, focusing on the case where S in finitely generated (where C(S) is known to be finite). Answering an open question posed by Garćıa-Sánchez, we provide a method to compute the smallest nonzero element of C(S) that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for C(S), and present some open questions for further study.